The problem of interest is to understand how both temperature and the rate of heat loss in the contents of the tank will vary as a consequence of unplanned and possibly random changes in Tin, and operator-introduced actions in the rate of heat addition Q. The rate of heat loss is a function of volume V and tank temperature T, as defined by the equation, QL= αVT^2

α is a constant-valued coefficient. You may assume the following:

a. Consider the scenario where both inlet and outlet volumetric flow rates are equal, i. e., q_in = q_out = q.
b. Densities and heat capacities are constant and equal for all streams: pressure is constant through the process.
c. The system is well-mixed: furthermore, the rate of shaft work is negligible.

Please perform the following:

1. Use the appropriate conservation and accounting statements to obtain the system of equations that describe the dynamics of the problem. Clearly state your assumptions, system and system boundaries, and simplification steps required to obtain the appropriate model equations.
2. Represent the system of equations from part a) in lumped parameter state-space form, and indicate the inputs (manipulated and disturbances), outputs, and states.